4.2 The large deviation principles
By the exponential equivalence, Proposition 4.4, and by [Dembo and Zeitouni, 1998, Theorem 4.2.13] it suffices to prove the large and moderate deviation principles in the framework of the
exponentially equivalent processes 17 constructed in the previous section.
The first step in the proof of the first part of Theorem 1.13, is to show a large deviation principle for the occupation times of the underlying process. Throughout this section we denote
a
κ
:= κ
1 1−α
¯ ℓκ.
We define the function ξ: R → −∞, ∞] by
ξu = log
1 1
−u
if u 1,
∞ otherwise.
Its Legendre-Fenchel transform is easily seen to be ξ
∗
t = t
− 1 − log t if t
0, ∞
otherwise.
Lemma 4.5. For fixed 0 ¶ u v the family
1 κ
T [ κu, κv
κ0
satisfies a large deviation principle with speed a
κ
and rate function Λ
∗ u,v
t = sup
ζ∈R
[tζ − Λ
u,v
ζ], where Λ
u,v
ζ = Z
v u
s
α 1
−α
ξζs
−α1−α
ds.
Proof. For fixed u v denote by I
κ
= I
[u,v κ
= { j ∈ N ∪ {0}: Φ j ∈ [κu, κv}. We get, using S
j
for the underlying sequence of Exp f j-distributed independent random variables,
Λ
κ
θ := log Ee
θ T [κu,κvκ
= X
j ∈I
κ
log Ee
θ κ
S
j
= X
j ∈I
κ
log 1
1 −
θ κ f j
= X
t ∈ΦI
κ
ξ θ
κ f Φ
−1
t =
Z
¯I
κ
f Φ
−1
t ξ θ
κ f Φ
−1
t d t,
where ¯I
κ
= ¯I
[u,v κ
= S
j ∈I
κ
[Φ j, Φ j + 1. Now choose θ in dependence on κ as θ
κ
= ζκ
1 1−α
¯ ℓκ
with ζ u
α1−α
. Then Z
¯I
κ
¯ f t
ξ θ
κ
κ ¯ f t
d t = κ
Z
¯I
κ
κ
¯ f
κsξ θ
κ
κ ¯ f
κs ds
= κ
1 1−α
Z
¯I
κ
κ
s
α 1
−α
¯ ℓκsξ
ζ¯ℓκ s
α 1
−α
¯ ℓκs
ds.
1245
Note that inf¯I
κ
κ and sup¯I
κ
κ approach the values u and v, respectively. Hence, we conclude with the dominated convergence theorem that one has
Λ
κ
θ
κ
∼ κ
1 1−α
¯ ℓκ
Z
v u
s
α 1
−α
ξ
ζ s
α 1
−α
ds |
{z }
=Λ
u,v
ζ
as κ tends to infinity. Now the Gärtner-Ellis theorem implies the large deviation principle for the
family T [ κu, κv
κ0
for 0 u v. It remains to prove the large deviation principle for u = 0.
Note that ET [0,
κv = E X
j ∈I
κ
S
j
= Z
¯I
κ
f Φ
−1
t 1
f Φ
−1
t d t
∼ κv and
varT [0, κv =
X
j ∈I
κ
varS
j
= Z
¯I
κ
f Φ
−1
t 1
f Φ
−1
t
2
d t ® 1
f 0 κv.
Consequently,
T [0, κǫ
κ
converges in probability to ǫ. Thus for t v
P
1 κ
T [0, κv ¶ t
¾ P
1 κ
T [0, κǫ ¶ 1 + ǫǫ
| {z
}
→1
P
1 κ
T [ κǫ, κv ¶ t − 1 + ǫǫ
and for sufficiently small ǫ 0
lim inf
κ→∞
1 a
κ
log P
1 κ
T [0, κv ¶ t
¾ −Λ
∗ ǫ,v
t − 1 + ǫǫ, while the upper bound is obvious.
The next lemma is necessary for the analysis of the rate function in Lemma 4.5. It involves the
function ψ defined as ψt = 1 − t + t log t for t ¾ 0.
Lemma 4.6. For fixed 0
x x
1
there exists an increasing function η: R
+
→ R
+
with lim
δ↓0
η
δ
= 0 such that for any u, v
∈ [x , x
1
] with δ := v − u 0 and all w ∈ [u, v], t 0 one has Λ
∗ u,v
t − w
α 1
−α
t ψ
δ t
¶ η
δ
δ + t ψ
δ t
. We now extend the definition of Λ
∗
continuously by setting, for any u ¾ 0 and t ¾ 0, Λ
∗ u,u
t = u
α 1
−α
t. For the proof of Lemma 4.6 we use the following fact, which can be verified easily.
Lemma 4.7. For any ζ 0 and t 0, we have |ξ
∗
ζt − ξ
∗
t| ¶ 2|ζ − 1| + | log ζ| + 2|ζ − 1|ξ
∗
t.
Proof of Lemma 4.6. First observe that
γ
δ
:= sup
x uvx
1
v −u ¶ δ
vu
α 1
−α
1246
tends to 1 as δ tends to zero. By Lemma 4.7, there exists a function ¯
η
δ δ0
with lim
δ↓0
¯ η
δ
= 0 such that for all
ζ ∈ [1γ
δ
, γ
δ
] and t 0 |ξ
∗
ζt − ξ
∗
t| ¶ ¯ η
δ
1 + ξ
∗
t. Consequently, one has for any
δ 0, x w, ¯
w x
1
with |w − ¯
w | ¶ δ and ζ ∈ [1γ
δ
, γ
δ
] that | ¯
w
α 1
−α
ξ
∗
ζt − w
α 1
−α
ξ
∗
t| ¶ ¯ w
α 1
−α
|ξ
∗
ζt − ξ
∗
t| + ξ
∗
t| ¯ w
α 1
−α
− w
α 1
−α
| ¶ c ¯
η
δ
1 + ξ
∗
t + cδξ
∗
t, where c
∞ is a constant only depending on x , x
1
and α. Thus for an appropriate function η
δ δ0
with lim
δ↓0
η
δ
= 0 one gets | ¯
w
α 1
−α
ξ
∗
ζt − w
α 1
−α
ξ
∗
t| ¶ η
δ
1 + ξ
∗
t. 18
Fix x u v x
1
and set δ := v − u. We estimate, for θ ¾ 0,
δu
α 1
−α
ξθ v
−α1−α
¶ Λ
u,v
θ ¶ δv
α 1
−α
ξθ u
−α1−α
, and the reversed inequalities for
θ ¶ 0. Consequently, Λ
∗ u,v
δt = sup
θ
[θ t − Λ
u,v
θ ] ¶
δ sup
θ
[θ t − u
α 1
−α
ξθ v
−α1−α
] ∨ δ sup
θ
[θ t − v
α 1
−α
ξθ u
−α1−α
] = δu
α 1
−α
ξ
∗
vu
α 1
−α
t ∨ δv
α 1
−α
ξ
∗
uv
α 1
−α
t. Since v
u
α1−α
and u v
α1−α
lie in [1 γ
δ
, γ
δ
] we conclude with 18 that for w ∈ [u, v Λ
∗ u,v
δt ¶ w
α 1
−α
ξ
∗
tδ + η
δ
1 + ξ
∗
tδ. To prove the converse inequality, observe
Λ
∗ u,v
t ¾ δ sup
θ ¶0
[θ t − u
α 1
−α
ξθ v
−α1−α
] ∨
δ sup
θ ¾0
[θ t − v
α 1
−α
ξθ u
−α1−α
] .
Now note that the first partial Legendre transform can be replaced by the full Legendre transform if t ¶ u
v
α1−α
. Analogously, the second partial Legendre transform can be replaced if t ¾ vu
α1−α
. Thus we can proceed as above if t 6∈ 1γ
δ
, γ
δ
and conclude that Λ
∗ u,v
t ¾ w
α 1
−α
ξ
∗
tδ − η
δ
1 + ξ
∗
tδ. The latter estimate remains valid on 1
γ
δ
, γ
δ
if x
α1−α 1
ξ
∗
1γ
δ
∨ξ
∗
γ
δ
¶ η
δ
. Since γ
δ
tends to 1 and
ξ
∗
1 = 0 one can make η
δ
a bit larger to ensure that the latter estimate is valid and lim
δ↓0
η
δ
= 0. This establishes the statement.
As the next step in the proof of Theorem 1.13 we formulate a finite-dimensional large deviation principle, which can be derived from Lemma 4.5.
1247
Lemma 4.8. Fix 0 = t t
1
· · · t
p
. Then the vector
1 κ
Z
κt
j
: j ∈ {1, . . . , p}
satisfies a large deviation principle in {0 ¶ a
1
¶ · · · ¶ a
p
} ⊂ R
p
with speed a
κ
and rate function J a
1
, . . . , a
p
=
p
X
j=1
Λ
∗ a
j −1
,a
j
t
j
− t
j −1
, with a
:= 0 .
Proof. First fix 0 = a a
1
· · · a
p
. Observe that, whenever s
j −1
s
j
with s = 0,
P
1 κ
Z
κt
j
¾ a
j 1
κ
Z
κs
j
for j ∈ {1, . . . , p}
¾ P s
j
− s
j −1
1 κ
T [a
j −1
κ, a
j
κ ¶ t
j
− t
j −1
for j ∈ {1, . . . , p}
. Moreover, supposing that 0
t
j
− t
j −1
− s
j
− s
j −1
¶ δ for a δ 0, we obtain P a
j
¶
1 κ
Z
κt
j
a
j
+ ǫ for j ∈ {1, . . . , p} ¾ P
1 κ
Z
κt
j
¾ a
j 1
κ
Z
κs
j
and T [a
j
κ, a
j
+ ǫκ ¾ δ for j ∈ {1, . . . , p} By Lemma 4.5, given
ǫ 0 and A 0, we find δ 0 such that, for κ large, P
1 κ
T [a
j
κ, a
j
+ ǫκ δ ¶ e
−Aa
κ
. Hence, for sufficiently small
δ we get with the above estimates that lim inf
κ→∞
1 a
κ
log P a
j
+ ǫ
1 κ
Z
κt
j
¾ a
j
for j ∈ {1, . . . , p}
¾ lim inf
κ→∞
1 a
κ
log P s
j
− s
j −1
1 κ
T [a
j −1
κ, a
j
κ ¶ t
j
− t
j −1
for j ∈ {1, . . . , p}
¾ −
p
X
j=1
Λ
∗ a
j −1
,a
j
t
j
− t
j −1
. Next, we prove the upper bound. Fix 0 = a
¶ . . . ¶ a
p
and 0 = b ¶ . . . ¶ b
p
with a
j
b
j
, and observe that by the strong Markov property of Z
t
, P b
j 1
κ
Z
κt
j
¾ a
j
for j ∈ {1, . . . , p}
=
p
Y
j=1
P b
j 1
κ
Z
κt
j
¾ a
j
b
i 1
κ
Z
κt
i
¾ a
i
for i ∈ {1, . . . , j − 1}
¶
p
Y
j=1
P
1 κ
T [b
j −1
κ, a
j
κ t
j
− t
j −1
¶
1 κ
T [a
j −1
κ, b
j
κ .
Consequently, lim sup
κ↑∞
1 a
κ
log P b
j 1
κ
Z
κt
j
¾ a
j
for j ∈ {1, . . . , p}
¶ −
p
X
j=1
r
j
, 1248
where r
j
=
Λ
∗ b
j −1
,a
j
t
j
− t
j −1
if a
j
− b
j −1
¾ t
j
− t
j −1
, Λ
∗ a
j −1
,b
j
t
j
− t
j −1
if b
j
− a
j −1
¶ t
j
− t
j −1
, 0,
otherwise. Using the continuity of u, v
7→ Λ
∗ u,v
t for fixed t, it is easy to verify continuity of each r
j
of the parameters a
j −1
, a
j
, b
j −1
, and b
j
. Suppose now that a
j
and b
j
are taken from a predefined compact subset of R
d
. Then we have
p
X
j=1
r
j
− Λ
∗ a
j −1
,a
j
t
j
− t
j −1
¶ ϑ max{b
j
− a
j
: j = 1, . . . , p }
, for an appropriate function
ϑ with lim
δ↓0
ϑδ = 0... Now the upper bound follows with an obvious exponential tightness argument.
We can now prove a large deviation principle in a weaker topology, by taking a projective limit and
simplifying the resulting rate function with the help of Lemma 4.6.
Lemma 4.9. On the space of increasing functions with the topology of pointwise convergence the family of functions
1 κ
Z
κt
: t ¾ 0
κ0
satisfies a large deviation principle with speed a
κ
and rate function J .
Proof. Observe that the space of increasing functions equipped with the topology of pointwise convergence can be interpreted as projective limit of the spaces
{0 ¶ a
1
¶ · · · ¶ a
p
} with the canonical projections given by
πx = xt
1
, . . . , xt
p
for 0 t
1
. . . t
p
. By the Dawson- Gärtner theorem, we obtain a large deviation principle with good rate function
˜ J x =
sup
t
1
...t
p
p
X
j=1
Λ
∗ xt
j −1
,xt
j
t
j
− t
j −1
. Note that the value of the variational expression is nondecreasing, if additional points are added to
the partition. It is not hard to see that ˜ J x =
∞, if x fails to be absolutely continuous. Indeed, there exists
δ 0 and, for every n ∈ N, a partition δ ¶ s
n 1
t
n 1
¶ · · · ¶ s
n n
t
n n
¶
1 δ
such that
P
n j=1
t
n j
− s
n j
→ 0 but P
n j=1
xt
n j
− xs
n j
¾ δ. Then, for any λ 0, ˜
J x = sup
t1...tp λ1,...,λp∈R
p
X
j=1
λ
j
t
j
− t
j −1
− Λ
xt
j −1
,xt
j
λ
j
¾
n
X
j=1
h − λ t
n j
− s
n j
+ Z
xt
n j
xs
n j
u
α 1
−α
log 1 + λu
−α 1
−α
du i
¾ −λ
n
X
j=1
t
n j
− s
n j
+ δ
1 1
−α
log 1 + λδ
α 1
−α
−→ δ
1 1
−α
log 1 + λδ
α 1
−α
,
1249
which can be made arbitrarily large by choice of λ.
From now on suppose that x is absolutely continuous. The remaining proof is based on the equation ˜
J x = sup
t
1
...t
p
p
X
j=1
t
j
− t
j −1
xt
j
α 1
−α
ψ xt
j
− xt
j −1
t
j
− t
j −1
. 19
Before we prove its validity we apply 19 to derive the assertions of the lemma. For the lower bound we choose a scheme 0
t
n 1
· · · t
p n
, with p depending on n, such that t
p n
→ ∞ and the mesh goes to zero. Define, for t
n j
−1
¶ t t
n j
, x
n j
t = 1
t
n j
− t
n j
−1
Z
t
n j
t
n j
−1
˙ x
s
ds = xt
n j
− xt
n j
−1
t
n j
− t
n j
−1
. Note that, by Lebesgue’s theorem, x
n j
t → ˙x
t
almost everywhere. Hence ˜
J x ¾ lim inf
n →∞
Z
t
n p
x
α 1
−α
t
ψ x
n j
t d t ¾
Z
∞
x
α 1
−α
t
lim inf
n →∞
ψ x
n j
t d t = J x.
For the upper bound we use the convexity of ψ to obtain
ψ xt
j
− xt
j −1
t
j
− t
j −1
= ψ 1
t
j
− t
j −1
Z
t
j
t
j −1
˙ x
t
d t ¶
1 t
j
− t
j −1
Z
t
j
t
j −1
ψ ˙ x
t
d t. Hence
˜ J x ¶
sup
t
1
...t
p
p
X
j=1
xt
j
α 1
−α
Z
t
j
t
j −1
ψ ˙ x
t
d t = J x , as required to complete the proof.
It remains to prove 19. We fix t
′
and t
′′
with t
′
t
′′
and xt
′
0, and partitions t
′
= t
n
· · · t
n n
= t
′′
with δ
n
:= sup
j
xt
n j
− xt
n j
−1
converging to 0. Assume n is sufficiently large such that η
δ
n
¶
1 2
t
′
α 1
−α
, with η as in Lemma 4.6. Then,
n
X
j=1
Λ
∗ xt
n j
−1
,xt
n j
t
n j
− t
n j
−1
¾ 1
2 t
′
α 1
−α
h
n
X
j=1
t
n j
− t
n j
−1
ψ xt
n j
− xt
n j
−1
t
n j
− t
n j
−1
| {z
}
∗
−xt
′′
− xt
′
i ,
20
and ∗ is uniformly bounded as long as ˜
J x is finite. On the other hand also the finiteness of the right hand side of 19 implies uniform boundedness of
∗. Hence, either both expressions in 19
1250
are infinite or we conclude with Lemma 4.6 that for an appropriate choice of t
n j
, sup
t
′
=t ···t
p
=t
′′
p
X
j=1
Λ
∗ xt
j −1
,xt
j
t
j
− t
j −1
= lim
n →∞
n
X
j=1
Λ
∗ xt
n j
−1
,xt
n j
t
n j
− t
n j
−1
= lim
n →∞
n
X
j=1
t
n j
− t
n j
−1
xt
n j
α 1
−α
ψ xt
n j
− xt
n j
−1
t
n j
− t
n j
−1
= sup
t
′
=t ···t
p
=t
′′
p
X
j=1
t
j
− t
j −1
xt
j
α 1
−α
ψ xt
j
− xt
j −1
t
j
− t
j −1
. This expression easily extends to formula 19.
Lemma 4.10. The level sets of J are compact in
I [0, ∞.
Proof. We have to verify the assumptions of the Arzelà-Ascoli theorem. Fix δ ∈ 0, 1, t ¾ 0, and
a function x ∈ I [0, ∞ with finite rate J. We choose δ
′
∈ 0, δ with x
t+ δ
′
=
1 2
x
t
+ x
t+ δ
, denote ǫ = x
t+ δ
− x
t
, and observe that J x ¾
Z
t+ δ
t
x
α 1
−α
s
[1 − ˙x
s
+ ˙ x
s
log ˙ x
s
] ds ¾
δ − δ
′
ǫ 2
α 1
−α
Z
t+ δ
t+ δ
′
[1 − ˙x
s
+ ˙ x
s
log ˙ x
s
] ds
δ − δ
′
. Here we used that x
s
¾ ǫ2 for s ∈ [t + δ
′
, t + δ]. Next, we apply Jensen’s inequality to the convex
function ψ to deduce that
J x ¾ δ − δ
′
ǫ 2
α 1
−α
ψ 1
δ − δ
′
ǫ 2
. Now assume that
ǫ 2
¾ δ. Elementary calculus yields
J x ¾ δ
ǫ 2
α 1
−α
ψ 1
δ ǫ
2 ¾
ǫ 2
1 1
−α
log ǫ
2e δ
. If we additionally assume
ǫ ¾ 2eδ
1 2
, then we get J x log δ
−
1 2
1 −α
¾ ǫ. Therefore, in general
x
t+ δ
− x
t
¶ max 2
J x log
δ
−
1 2
1 −α
, 2e δ
1 2
. Hence the level sets are uniformly equicontinuous. As x
= 0 for all x ∈ I [0, ∞ this implies that the level sets are uniformly bounded on compact sets, which finishes the proof.
We now improve our large deviation principle to the topology of uniform convergence on compact
sets, which is stronger than the topology of pointwise convergence. To this end we introduce, for every m
∈ N, a mapping f
m
acting on functions x : [0, ∞ → R by
f
m
x
t
= x
t
j
if t
j
:=
j m
¶ t
j+1 m
=: t
j+1
. 21
1251
Lemma 4.11. For every δ 0 and T 0, we have
lim
m →∞
lim sup
κ↑∞
1 a
κ
log P sup
0¶t¶T
f
m 1
κ
Z
κ · t
−
1 κ
Z
κt
δ = −∞.
Proof. Note that
P sup
0¶t¶T
f
m 1
κ
Z
κ · t
−
1 κ
Z
κt
¾ δ
¶
T m
X
j=0
P
1 κ
Z
κt
j+1
−
1 κ
Z
κt
j
¾ δ
. By Lemma 4.9 we have
lim sup
κ↑∞
1 a
κ
log P
1 κ
Z
κt
j+1
−
1 κ
Z
κt
j
¾ δ
¶ inf J x: x
t
j+1
− x
t
j
¾ δ
, and, by Lemma 4.10, the right hand side diverges to infinity, uniformly in j, as m
↑ ∞.
Proof of the first large deviation principle in Theorem 1.13. We apply [Dembo and Zeitouni,
1998, Theorem 4.2.23], which allows to transfer the large deviation principle from the topolog- ical Hausdorff space of increasing functions with the topology of pointwise convergence, to the
metrizable space I [0, ∞ by means of the sequence f
m
of continuous mappings approximating the identity. Two conditions need to be checked: On the one hand, using the equicontinuity of the sets
{Ix ¶ η} established in Lemma 4.10, we easily obtain lim sup
m →∞
sup
J x ¶ η
d f
m
x, x = 0,
for every η 0, where d denotes a suitable metric on I [0, ∞. On the other hand, by Lemma 4.11,
we have that f
m 1
κ
Z
κ ·
are a family of exponentially good approximations of
1 κ
Z
κ ·
.
The proof of the second large principle can be done from first principles.
Proof of the second large deviation principle in Theorem 1.13. For the lower bound observe
that, for any T 0 and ǫ 0,
P sup
0¶t¶T
|
1 κ
Z
κt
− t − a
+
| ǫ ¾ P
sup
0¶t¶T
|
1 κ
Z
κt
− t − a
+
| ǫ, Z
κa
= 0 ¾ P Z
κa
= 0 P
sup
a¶t¶T
|
1 κ
Z
κt
− Z
κa
− t − a| ǫ ,
and recall that the first probability on the right hand side is exp {−κ a f 0} and the second converges
to one, by the law of large numbers. For the upper bound note first that, by the first large deviation principle, for any
ǫ 0 and closed set A ⊂ {Jx ǫ}, lim sup
κ↑∞
1 κ
log P
1 κ
Z
κ ·
∈ A = −∞.
1252
Note further that, for any δ 0 and T 0, there exists ǫ 0 such that Jx ¶ ǫ implies
sup
0¶t¶T
|x − y| δ, where y
t
= t − a
+
for some a ∈ [0, T ]... Then, for θ f 0,
P sup
0¶t¶T
|
1 κ
Z
κt
− y| ¶ δ ¶ P Z
κa
¶ δκ
= P T [0, κδ] ¾ κa ¶ e
−κaθ
Y
Φ j¶κδ
E exp θ S
j
= e
−κaθ
exp X
Φ j¶κδ
log 1
1 −
θ f j
, and the result follows because the sum on the right is bounded by a constant multiple of
κδ.
4.3 The moderate deviation principle
Kategori